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Journal of the London Mathematical Society 1991 s2-43(3):545-555; doi:10.1112/jlms/s2-43.3.545
© 1991 by London Mathematical Society
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© Oxford University Press

Elliptic Hopf Algebras

Yves Félix, Stephen Halperin and Jean-Claude Thomas

Institut de Mathématiques, Université Catholique de Louvain B1348 Louvain-la-Neuve, Belgium
Physical Sciences Divisio, Scarborough College, University of Toronto Scarborough, Canada M1C 1A4
U.F.R. de Mathématiques, Université de Lille Flandres, Artois, 59655 Villeneuve d'Ascq, France

An elliptic Hopf algebra is a connected graded cocommutative Hopf algebra that is finitely generated and nilpotent. If (A, m, k) is a local noetherian ring then ExtA(k; k) is elliptic if and only if A is a complete intersection. Similarly, special conditions are imposed on a simply connected topological space X if H*({Omega}X; k) is elliptic.

Elliptic Hopf algebras G have finite depth and we show that they are characterized among Hopf algebras of finite depth by any of the following three properties: (i) {sum}i≤n dim Gi grows at most polynomially in n; (ii) G is left noetherian; (iii) G is nilpotent.


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